Haibo Chen

Existence of positive solutions for nonlinear fractional functional differential equation

In this paper, the existence of positive solutions for the nonlinear Caputo fractional functional differential equation in the form {D0+^qy(t)+r(t)f(yt)=0,@?t@?(0,1),q@?(n-1,n],y^(^i^)(0)=0,0@?i@?n-3,@ay^(^n^-^2^)(t)-@by^(^n^-^1^)(t)=@h(t),t@?[-@t,0],@cy^(^n^-^2^)(t)+@dy^(^n^-^1^)(t)=@x(t),t@?[1,1+a] is studied. By constructing a special cone and using Krasnoselskiis fixed point theorem, various results on the existence of at least one or two positive solutions to the fractional functional differential equation are established. The main results improve and generalize the existing results.

Stability and bifurcations of limit cycles of the equator in a class of cubic polynomial systems

Abstract In this paper, we study the appearance of limit cycles from the equator in a class of cubic polynomial vector fields with no singular points at infinity and the stability of the equator of the systems. We start by deducing the recursion formula for quantities at infinity in these systems, then specialize to a particular case of these cubic systems for which we study the bifurcation of limit cycles from the equator. We compute the quantities at infinity with computer algebraic system Mathematica 2.2 and reach with relative ease an expression of the first six quantities at infinity of the system, and give a cubic system, which allows the appearance of six limit cycles in the neighborhood of the equator. As far as we know, this is the first time that an example of cubic system with six limit cycles bifurcating from the equator is given. The technique employed in this work is essentially different from more usual ones. The recursion formula we present in this paper for the calculation of quantities at infinity is linear and then avoids complex integrating operations. Therefore, the calculation can be readily done with using computer symbol operation system such as Mathematica.

Multiple solutions for superlinear Schrödinger-Poisson system with sign-changing potential and nonlinearity

In this paper, we study the following Schrodinger-Poisson system { - Δ u + V ( x ) u + λ ? ( x ) u = f ( x , u ) , in? R 3 , - Δ ? = u 2 , in? R 3 , where the nonlinearity f and the potential V are allowed to be sign-changing. Under some appropriate assumptions on V and f , we obtain the existence of two different solutions of the system via the Ekelands variational principle and the Mountain Pass Theorem.

Generalized quasilinear asymptotically periodic Schrödinger equations with critical growth

This paper is concerned with the following quasilinear Schrodinger equations: { - div ( g 2 ( u ) ? u ) + g ( u ) g ( u ) | ? u | 2 + V ( x ) u = λ f ( x , u ) + g ( u ) | G ( u ) | 2 ? - 2 G ( u ) , x ? R N , u ( x ) 0 , x ? R N , where N ? 3 , 2 ? = 2 N N - 2 , G ( u ) = ? 0 u g ( t ) d t and λ 0 is a parameter. By using a change of variable, the quasilinear equation is reduced to a semilinear one, whose associated functional is well defined in H 1 ( R N ) . We establish the existence of positive solutions for this problem by using the Mountain Pass Theorem in combination with the concentration-compactness principle under appropriate assumptions on V ( x ) and f ( x , u ) . Recent results from the literature are improved and extended.

Multiple positive solutions of n-point boundary value problems for p-Laplacian impulsive dynamic equations on time scales

In this paper, we study the n-point boundary value problems for p-Laplacian impulsive dynamic equations on time scales. By using the Leray-Schauder fixed point theorem and the nonlinear alternative of Leray-Schauder type, we get the existence of at least one positive solution. We also consider the existence of at least three positive solutions by using a new fixed point theorem. As an application, we give an example to demonstrate our results.

Three-point boundary value problems for second-order ordinary differential equations in Banach spaces

In this paper, by using the Sadovskii fixed point theorem, we study the existence of at least one solution for the second-order three-point boundary value problem u^(t)+f(t,u(t),u^(t))[emailxa0protected],0

Multiple solutions for a nonlinear Schrödinger-Poisson system with sign-changing potential

This paper deals with the existence and multiplicity of nontrivial solutions for a nonlinear Schrodinger-Poisson system. Under some suitable conditions, some criteria on the existence of nontrivial solutions, including a ground state solution, two solutions and a sequence of nontrivial solutions { u k } with ? u k ? ? 0 as k ? + ∞ , is established by applying the Morse theory and variational methods. Some known results in the literatures are extended and improved.

Variational methods to the second-order impulsive differential equation with Dirichlet boundary value problem

In this paper, we study the existence of solutions for a class of second-order impulsive differential equation. By using the critical point theorem of Y. Jabri and an even functional theorem, we give some new criteria to guarantee that the impulsive differential equation has at least one solution, infinitely many solutions under the assumption that a nonlinear term satisfies sublinear, superlinear, asymptotically linear, respectively. Some recent results are extended and conditions of assumptions are simplified. Finally, some examples are presented to illustrate our main results.

Triple positive solutions for nonlinear boundary value problems in Banach space

In this paper, by applying two pairs of lower and upper solutions method and the topological degree theory of strict-set-contractions, the existence of at least three positive solutions to m-point boundary value problems for second order ordinary differential equations in Banach spaces is obtained.

Calculation of singular point quantities at infinity for a type of polynomial differential systems

The center problem at infinity is far to be solved in general. In this paper we develop a procedure to resolve it for a particular type of polynomial differential systems. The problem is solved by writing its concomitant differential equation in the complex coordinates introduced by Yirong Liu and by developing a new method of computation of the so called singular point quantities. This method is based on the transformation of infinity into the elementary origin. Finally, the investigation of center problem for the infinity of a particular family of planar polynomial vector fields of degree 5 is carried out to illustrate the main theoretical results. These involve extensive use of a Computer Algebra System, we have chosen to use Mathematica?.